"This is the kind of story that gives me nightmares... because I can see myself making the same mistake."
"The fundamental group in general is the set of topologically equivalent maps from S1 into your space X."
"The other main idea behind covariance is kind of a bummer: covariance in and of itself is not very interesting."
"Negative areas, which make no sense in reality, must exist as an intermediate step on the way to the solution."
"It is, in fact, a fractal—arguably the most famous fractal is the Mandelbrot set."
"I'm not saying a time period. I'm saying a step function."
"If we know a periodic function on an interval of length 1, we know it everywhere."
"Now since the infinite expressions are equal in a way, by expanding the infinite product we expect to recover the infinite sum."
"The logarithms can simplify even the most complicated expressions."
"Category theory provides a language to help you identify when two things are really the same."
"If I take 30 steps linearly, I get to 30. If I take 30 steps exponentially, I get to a billion." - Ray Kurzweil
"What the computer is doing instead is thinking about elliptic curves modulo primes."
"Remember the fraction, the decimal, and the percentage."
"A structure that repeats itself on smaller and smaller scales is called a fractal."
"It's really important to know what type of matrices are you getting back."
"Gradient descent in particular has a slower rate. It's one over k."
"Think of it as a conceptual shorthand for 'the best constant approximation for the rate of change'."
"There's a nice little thing that you can define - it's a finite number."
"Think of it as a calculus... for calculating probabilities."
"Negative numbers are funny in that sometimes they make sense sometimes they don't."
"Power laws are scale-free, there's no special size, no favored size, and mathematically there's no mean."
"So whenever we see the derivative of one of these sigmoids with respect to its input, we can just write the output times one minus alpha, and we've got it."
"Just knowing your basic vector arithmetic dot product and cross product, that's going to give you guys 99% of what you need in this industry to succeed."
"He knew the Euler number and encrypted it in the most famous piece of art in the world."
"Find the GCF by identifying the highest common factor between the given numbers or terms."
"So we actually know only six terms and the sixth is 60."
"Imaginary numbers do exist, and they have their place in understanding the universe."
"The epsilon delta limit is almost like an atom for calculus and in that way it captures the whole idea of calculus in itself."
"Some infinities are literally bigger than other infinities."
"If you can reduce the golf ball paradox down to a ball bouncing around the inside of a four-sided shape, why not go further?"
"The golden ratio... it can be defined by taking a line and breaking it into two separate pieces."
"The inflection point occurs when the slope stops increasing."
"Factoring comes up over and over and over again in algebra."
"You're not going to have a complete toolkit factoring toolkit if you don't understand the difference of two squares."
"Graph theory is all about the study of the properties of these types of networks, and how they can be used to model and solve a whole host of interesting problems."
"A cycle is defined as a path that starts and ends at the same vertex."
"We're talking about probability, not about possible and impossible."
"To use them properly, we just have to remember that they don't care which way is left and which is right."
"The heart of calculus: the strategic use of infinity to solve hard problems."
"Calculus has a double-barreled strategy-- cut and then rebuild."
"If you think of the cone going both ways you get both parts of the hyperbola."
"The first problem calculus solves for us is the area problem."
"The second problem calculus solves for us is something called the rate of change."
"Thinking about this in the context of a distribution, if we had a distribution with some very narrow width, if this width gets extraordinarily narrow, then no matter what F of X does out here, we don't care."
"We're representing our function as a sum of sine waves. This is the basic idea of a Fourier series."
"The sum of probabilities for mutually exclusive outcomes would be one."
"Every number x on the real number line has one of these star-friends, the number x plus star, which is 'confused' with x, but not equal to it."
"If a random variable X has a cumulative density function F of X, then the variable F inverse of U, where U is a random uniform variable between 0 and 1, also has CDF F of X."
"A continuous random variable takes any value along a given interval of a number line."
"Algebra 1 is really not much more than variables, expressions, graphs, and rules."
"This concept of a limit is the very first thing you learn in calculus one."
"The conception of the complex plane was no easy feat."
"The Fourier transform of the constant one seems to be a delta function at zero of area 2pi."
"You let Excel handle the math for you, and that lets you as the HR professional focus on the concepts."
"There's a difference between simplifying and rationalizing."
"...the transition from Fourier series to the Fourier transform."
"Any decimal that stops can always be written as a fraction, and so they're all rational."
"The intuitive concept of continuity is that you haven't snipped anything, you haven't put any holes in anything; you've just twisted and stretched."
"It's this point of view that is so ubiquitous, not only in Fourier Series but in other versions of essentially the same ideas."
"Generating functions were a wonderful way to approach it."
"The distance between two points in Euclidean space is the sum of the distances between each of the components squared to the one-half."
"The Taylor series allows me to approximate nearby values in terms of these quantities."
"Functions are infinite dimensional in infinite dimensional vector space; they have an inner product so we can tell how close two functions are."
"Compactness really doesn't depend on the metric space that you're in."
"There's a formula for that, it's H equals to negative B over 2a."
"A manifold is a topological space where if you look at any point on it, that point has some little neighborhood that looks like Euclidean space."
"Convergent sequences' subsequences also converge and to the same limit."
"The interesting answer from Minkowski is that you can actually do a form of algebra on shapes."
"Anytime you have a step function, then the Fourier Series has a well, it does its best but it will overshoot by an amount that Gibbs found."
"One of the most important concepts in analysis is the concept of a continuous function."
"We want to find spaces that are locally Euclidean."
"For it to be true... it sounds like you can count to infinity."
"The mode is the highest frequency piece of data, the median is the middle value when we put them into ascending order, and the mean is when we calculate all of them added up divided by how many there are."
"Drawing graphs of functions tells you about the very nature of functions."
"The double integral represents the volume below the surface f of X Y but above the region R."
"A curved two-dimensional surface is a reasonable prototype for Riemannian geometry in a positive sense."
"Learn what stable and unstable mean, be able to get a picture of your general solution without even having to do work, that's pretty cool."
"These two topological spaces are the same, so we would call these two homeomorphic."
"A homeomorphism is an isomorphism in the category of topological spaces."
"This set is an orthogonal set of functions."
"The truncated Fourier series assumes the value midway between the two discontinuities; however, just before the discontinuity, there is an overshoot, and just after the discontinuity, there is an undershoot."
"Each of these types of minimization problems with greater than or equal problem constraints has an associated maximization problem that we call the dual problem."